I don't see why you'd need to increase the number of spaces on the board (other than obviously raising the power of .

Remember, in a 3D chessboard there'd already be 8*8*8 = 512 cubes on the board - and half of these would be allocated to the players, 128 each.

Alternatively one could have four-player chess with four lots of 32 cubes in the "corners" of the board. Though this would obviously involve 3 rows of 8 pawns and the original 8 pieces, which wouldn't fit your extra piece per dimension. Also, pawns would need to be able to move in multiple directions.

If we stick with my first idea, one possibility is to only have the center 16 cubes used for non-pawns, with the outer 48 cubes of the non-pawn layer empty. This is exactly double the number of non-pawn pieces of a 2D chessboard, which could be in the following quantities:64 pawns4 rooks (move out of the 6 faces of a cube)4 knights (move 2 cubes in one direction, one cube in all other directions)4 bishops (move out of the 8 vertices of a cube)2 of your new piece type (move out of the 12 edges of a cube)1 queen1 king

The only problem here is that in order to preserve symmetry the 4 knights would all have to go on one color of square and the 4 bishops on the other.

Alternatively if we really do have 64 non-pawn pieces (which does seem overkill), they could be like this:64 pawns16 rooks (at positions 0,0, 0,1 and 2,2, reflected around the layer in full symmetry)16 knights (at positions 0,2 and 2,3, reflected around the layer in full symmetry)16 bishops (at positions 0,3 and 1,2, reflected around the layer in full symmetry)12 of your new piece type (at positions 1,1 and 1,3, reflected around the layer in full symmetry)3 queens (they would probably have to have a different name !)1 king

The only remaining problem is that in either situation there would be at least two of some pieces which both start on the same color cube (light or dark) so you couldn't refer to them as "the light bishop" or "the dark rook" etc.

if the knight makes 2 squares one way and 1 another it's move remains in a 2D plane so I generalised it.(1 sq one way, 2 sq another way, 3 sq another way etc) the fact that the knight moves one extra square each time means extra squares are needed to it space to move. And one extra pair to accomadate the new piece. (different types of diagonal)That's why it needs extra spaces each time

It doesn't really matter though, there is no right or wrong way to generalise chess in higher dimensions I don't think(there's probably loads of ways)

It would be very weird to make the knight work like that. With my way the knight moves one space along all axes plus a further one space on one axis. Therefore it is always restricted to a hypercube of side 5 centred on the knight's current space. Also you wouldn't need a bigger board to accommodate the extra pieces as they are already way over-accounted for in the increase in dimensionality of the board.

I always loved the idea of Cheops (pyramid chess) in Dune. It would be a bit simpler than chess on a cube. Apparently the board has nine levels, which seems odd unless the bottom level is 9x9. Also there is a "dual objective": you have to get the queen to the top and the opponents king in check. I don't know if you have to do both or if the first person to do either wins. It would be hard to do both if both the queens died.

also, there is a possibility of modesty. it is already practiced. there are already spherical and toroidal chess that are being played. of course, it is not 4d chess, but the 2-dimensionality of the chess-board is taken one level up to 3d, so 8x8squares are folded into torus or sphere. most of these are played on magnetic boards, gravity has to be counted in when your queen is on the south pole or on the downside of the torus.